# Fused Entropy Algorithm in Optical Computed Tomography

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*Keywords:*optical computed tomography; fused entropy; reconstruction

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Key Laboratory of Space Active Opto-Electronics Technology, Shanghai Institute of Technical Physics of the Chinese Academy of Sciences, Shanghai 200083, China

Key laboratory of Nondestructive Test (Ministry of Education), Nanchang Hangkong University, Nanchang 330063, China

Author to whom correspondence should be addressed.

Received: 30 September 2013 / Revised: 10 February 2014 / Accepted: 10 February 2014 / Published: 17 February 2014

In most applications of optical computed tomography (OpCT), limited-view problems are often encountered, which can be solved to a certain extent with typical OpCT reconstructive algorithms. The concept of entropy first emerged in information theory has been introduced into OpCT algorithms, such as maximum entropy (ME) algorithms and cross entropy (CE) algorithms, which have demonstrated their superiority over traditional OpCT algorithms, yet have their own limitations. A fused entropy (FE) algorithm, which follows an optimized criterion combining self-adaptively ME with CE, is proposed and investigated by comparisons with ME, CE and some traditional OpCT algorithms. Reconstructed results of several physical models show this FE algorithm has a good convergence and can achieve better precision than other algorithms, which verifies the feasibility of FE as an approach of optimizing computation, not only for OpCT, but also for other image processing applications.

Optical computed tomography (OpCT) techniques such as interferometry tomography [1–3], light beam deflection tomography [4], emission spectral tomography [5–7], etc., are a branch of computed tomography (CT), which is mainly applied to optical testing of 3-D distributions of physical variables of a number of fluid fields [8–10]. Due to the limitations of the testing environments and devices, most OpCTs encounter limited-view problems [11–13], e.g., incomplete testing views and/or incomplete data at each view, which results in worse reconstruction precision and lower spatial resolution than that seen in medical CT. To solve this problem, many OpCT algorithms have been developed since Gordon et al. first proposed an algebraic reconstruction technique (ART) algorithm in the 1970s [14]. Gilbert introduced a simultaneous iterative reconstruction technique (SIRT) algorithm [15], in which each reconstructed pixel is revised after all the projection values have been computed in each iterative step. Aderson [16] proposed a simultaneous algebraic reconstruction technique (SART) algorithm, which combined the advantages of ART and SIRT. A natural pixel decomposition (NPD) algorithm was first proposed by Buonocore et al. [17], in which the shapes of the grids of the reconstructed plane are not rectangular but determined by the rays’ paths. Garnero et al. [18] employed the NPD to reconstruct a field of refractivity. Different from those above-mentioned row-relaxation iterative methods, a column-relaxation iterative reconstruction method was proposed by David [19]. Dean et al. [11] put forward a singular-value decomposition (SVD) algorithm that is suitable for the solution of both overdetermined and underdetermined equations. To solve the problem of the loss of projection data when reconstructing fields comprising obstacle objects, a discrete iterative reconstruction reprojection (DIRR) [20] was presented, which combines a low-pass filter with a re-projected estimation of the lost data. A Lagrange interpolation reprojection revising (LIRR) algorithm [21] adopts the pre-estimation of a Lagrange interpolation method to improve the accuracy of the re-projected estimation by DIRR, and has been demonstrated a rather improvement over DIRR.

OpCT algorithms are required to have accurate reconstruction results with incomplete data, which means optimization criteria have to be followed. Typical OpCT algorithms, such as ART, SIRT, SART, etc., mostly comply with a single optimization criterion, with which accurate reconstructed results can hardly be achieved. Although some multi-criterion OpCT algorithms [13] have been proposed, their reconstruction results are still unsatisfactory when the distribution of tested fields is relatively complex. Entropy, that first emerged in the information theory, has also been introduced into OpCT algorithms. Maximum entropy algorithms (ME) [22,23] search a most possible solution from the solution set by maximizing the entropy function of the tested target itself, which is verified as a superior approach compared to conventional OpCT algorithms when the number of views was extremely limited and tested fields are approximately symmetrical. However, the performance of ME degrades when the tested fields have poor symmetry. Cross entropy algorithms (CE) [24,25] figure out a most possible target function by minimizing the cross entropy function that measures the possible relationship between the distributions of the target function and its projections. The reconstruction precision of ME for asymmetrical targets can be improved when combined with CE [26].

In this paper, a fused entropy (FE) algorithm is proposed, which self-adaptively combines ME with CE, and hence has high reconstruction precision for both symmetrical and asymmetrical fields. The performance of FE is investigated by comparisons with ME, CE and some traditional OpCT algorithms in the reconstructions of several physical models. Results of numerical simulations show this FE has a good convergence and a better precision than other algorithms.

Like in medical CT, the projection data of fluid fields, which can be probed with optoelectronic sensors, are adopted to compute distributions of physical variables in the OpCT. From a mathematical point of view, the OpCT reconstruction problem can be formulated as the inverse Radon transform. As shown in Figure 1(a), the relationship between the 2-D physical function f (x, y) of a reconstructed plane of a tested fluid field and its projection q(t,θ) is given by:

$$q(t,\theta )={\int}_{-\infty}^{\infty}f(x,y)ds$$

The 2-D continuous function f (x, y) is normally discretized into grids for the OpCT reconstruction; M(N) even grids are discretized in the X(Y) directions, as shown in Figure 1(b). The discrete expression of the function f (x, y) is:

$$f(x,y)=\sum _{j=1}^{MN}{f}_{j}{b}_{j}=\sum _{j=1}^{MN}{f}_{j}b(x-{x}_{j},y-{y}_{j})$$

$$b(x-{x}_{j},y-{y}_{j})=b(\frac{x-m{l}_{x}}{{l}_{x}})b(\frac{y-n{l}_{y}}{{l}_{y}})$$

$$b(k)=\frac{\text{sin}k\pi}{k\pi}$$

Then we have:

$${q}_{i}(t,\theta )=\sum _{j=1}^{MN}{f}_{j}{\int}_{i}b(\frac{x-m{l}_{x}}{{l}_{x}})b(\frac{y-n{l}_{y}}{{l}_{y}})ds=\sum _{j=1}^{MN}{w}_{ij}{f}_{j}$$

$$\mathbf{Q}=\mathbf{WF}$$

Conventional OpCT algorithms usually follow a single optimization criterion. For example, the ART complies with a minimum norm criterion when a suitable initial image vector is selected and the SIRT subjects to a least-squares criterion.

Entropy concepts first based on information theory have also found application in OpCT algorithms. Maximum entropy (ME) algorithms maximize the entropy function of the reconstructed physical variables f_{j} (the image vector **F**):

$${\Phi}_{1}(\mathbf{F})=-\sum _{j=1}^{MN}{f}_{j}\text{ln}{f}_{j}=-{\mathbf{F}}^{T}\text{ln}\mathbf{F}$$

$${\Phi}_{2}(\mathbf{F})=\sum _{i=1}^{I}[(\sum _{j=1}^{MN}{w}_{ij}{f}_{j})\times \text{ln}(\frac{\sum _{j=1}^{MN}{w}_{ij}{f}_{j}}{{q}_{i}})]=\sum _{i=1}^{I}{\mathbf{W}}_{i}\mathbf{F}\times \text{ln}(\frac{{\mathbf{W}}_{i}\mathbf{F}}{{q}_{i}})$$

Here, we propose a fused entropy (FE) algorithm that self-adaptively combines ME with CE, which minimize:

$$\Phi (\mathbf{F})=-{\lambda}_{1}{\Phi}_{1}(\mathbf{F})+{\lambda}_{2}{\Phi}_{2}(\mathbf{F})$$

$$\frac{\text{d}\Phi (\mathbf{F})}{\mathbf{F}}={\lambda}_{1}(\text{ln}\mathbf{F}+\mathbf{1})+{\lambda}_{2}\sum _{i=1}^{I}\text{ln}(\frac{{\mathbf{W}}_{i}\mathbf{F}}{{q}_{i}}){\mathbf{W}}_{i}^{T}=\mathbf{0}$$

$$\begin{array}{l}{\mathbf{F}}^{0}=\mathbf{1}\\ {\mathbf{C}}^{k}=\mathbf{1}-\alpha [{\lambda}_{1}^{k}(\text{ln}{\mathbf{F}}^{\text{k}}+\mathbf{1})+{\lambda}_{2}^{k}\text{ln}(\frac{{\mathbf{W}}_{i}\mathbf{F}}{{q}_{i}}){\mathbf{W}}_{i}^{T}]\\ {\mathbf{F}}^{k+1}={\mathbf{C}}^{k}\cdot {\mathbf{F}}^{k}\end{array}$$

$$\begin{array}{l}{\lambda}_{1}^{0}={\lambda}_{2}^{0}=\frac{1}{2}\\ \{\begin{array}{l}{\lambda}_{1}^{k+1}\left|{\Phi}_{1}^{k+1}-{\Phi}_{1}^{k}\right|={\lambda}_{2}^{k+1}\left|{\Phi}_{2}^{k+1}-{\Phi}_{2}^{k}\right|\\ {\lambda}_{1}^{k+1}+{\lambda}_{2}^{k+1}=1\end{array}\end{array}$$

Several physical models including a top-concaved paraboloid image (TCP), a three random peaks image (TR), a superposition image (TCPTR) of the top-concaved paraboloid and the three random peaks, and a six-peak Gaussian image (SG), which are shown in Figure 2, are chosen to investigate the performance of FE algorithm.

TCP represents a complete symmetric distribution of physical variables and can be expressed as:

$${F}_{1}(x,y)=\{\begin{array}{rr}0,\hfill & \hfill {x}^{2}+{y}^{2}>{0.37}^{2}\\ 0.9[1-\frac{{x}^{2}}{{0.37}^{2}}-\frac{{y}^{2}}{{0.37}^{2}}],\hfill & \hfill {0.15}^{2}<{x}^{2}+{y}^{2}\le {0.37}^{2}\\ 0.9[1-\frac{{x}^{2}}{{0.37}^{2}}-\frac{{y}^{2}}{{0.37}^{2}}]-0.5[1-\frac{{x}^{2}}{{0.15}^{2}}-\frac{{y}^{2}}{{0.15}^{2}}],\hfill & \hfill {x}^{2}+{y}^{2}\le {0.15}^{2}\end{array}$$

$$\begin{array}{c}{F}_{2}(x,y)=\text{exp}\{-2.77[\frac{{(0.9x+0.2y+0.2)}^{2}}{{0.1}^{2}}+\frac{{(0.9\text{y}-0.2x)}^{2}}{{0.25}^{2}}]\}+\text{exp}\{-2.77[\frac{{(\frac{1}{2}x+\frac{\sqrt{3}}{2}y-0.2)}^{2}}{{0.11}^{2}}\\ +\frac{{(\frac{1}{2}y-\frac{\sqrt{3}}{2}x-0.1)}^{2}}{{0.3}^{2}}]\}+0.8\text{exp}\{-2.77[\frac{{(\frac{\sqrt{2}}{2}x+\frac{\sqrt{2}}{2}y)}^{2}}{{0.25}^{2}}+\frac{{(\frac{\sqrt{2}}{2}y-\frac{\sqrt{2}}{2}x+0.3)}^{2}}{{0.1}^{2}}]\}\end{array}$$

$${F}_{3}(x,y)={F}_{1}(x,y)+{F}_{2}(x,y)$$

$$\begin{array}{l}{F}_{4}(x,y)=\sum _{i=1}^{6}{a}_{i}\text{exp}\{[-\frac{4\text{ln}2}{{0.18}^{2}}[{(x-{x}_{i})}^{2}+{(y-{y}_{i})}^{2}]\}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{a}_{1}={a}_{4}=0.7,\hspace{0.17em}\hspace{0.17em}{a}_{2}={a}_{5}=1,\hspace{0.17em}\hspace{0.17em}{a}_{3}={a}_{6}=0.5\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{x}_{1}=0.3,\hspace{0.17em}\hspace{0.17em}{y}_{1}=0;\hspace{0.17em}\hspace{0.17em}{x}_{2}=0.3\text{cos}(\frac{\pi}{3}),\hspace{0.17em}\hspace{0.17em}{y}_{2}=0.3\text{sin}(\frac{\pi}{3});\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{x}_{3}=0.3\text{cos}(\frac{2\pi}{3}),\hspace{0.17em}\hspace{0.17em}{y}_{3}=0.3\text{sin}(\frac{2\pi}{3});\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{x}_{4}=-0.3,\hspace{0.17em}\hspace{0.17em}{y}_{4}=0;\hspace{0.17em}\hspace{0.17em}{x}_{5}=0.3\text{cos}(\frac{4\pi}{3}),\hspace{0.17em}\hspace{0.17em}{y}_{5}=0.3\text{sin}(\frac{4\pi}{3});\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{x}_{6}=0.3\text{cos}(\frac{5\pi}{3}),\hspace{0.17em}\hspace{0.17em}{y}_{6}=0.3\text{sin}(\frac{5\pi}{3})\end{array}$$

$${e}_{1}=(\sum _{j=1}^{MN}|{f}_{j}-{{f}_{j}}^{\prime}|)/({f}_{j\text{max}}\times MN)$$

$${e}_{2}={\left|{f}_{j}-{{f}_{j}}^{\prime}\right|}_{\text{max}}/{f}_{j\text{max}}$$

$${e}_{3}=\sqrt{\sum _{j=1}^{MN}{({f}_{j}-{{f}_{j}}^{\prime})}^{2}/\sum _{j=1}^{MN}{f}_{j}^{2}}$$

The performance of FE is investigated by comparisons with ME, CE and two traditional OpCT algorithms, i.e., the algebraic reconstruction technique (ART) algorithm and the simultaneous iterative reconstruction technique (SIRT) algorithm, in the reconstructions of the four physical models, where M = N = 256, RPV = 256, and V = 6 (evenly distributed views over the range of 180 degrees). The reconstruction errors of the five algorithms are shown in Table 1, where numbers in bold italic are the best results.

The four reconstructed images with the FE algorithm are shown in Figure 3, where the relaxation parameter α of FE is 0.3 for all the four physical models. Furthermore, the convergence of FE has been studied. Figure 4 shows the convergence properties of FE for the reconstructions of TCP and TR.

Numerical simulations show that the FE proposed in this paper is superior to the other four algorithms tested in the reconstructions of four physical models (refer to Table 1; note: only the average error is little greater than that of ME), and images reconstructed with FE are of similar distributions as the original physical models (refer to Figure 2 and Figure 3). Besides, FE also has a good convergence (refer to Figure 4). These studies have testified the feasibility of FE as an approach of optimizing computation, which can not only be utilized for OpCT reconstructions, but also be promoted to find an optimal solution for other imaging processing problems if an FE optimization function is set up [27]. However, owing to the complexity, such as turbulences and impulses in the fields of real OpCT applications, much deeper research needs to be conducted to verify the applicability of FE.

This work is jointly supported by Chinese Natural Science Fund under Grant 61271397, Jiangxi Natural Science Foundation under grant 20122BAB202009, Foundation of Jiangxi Education Bureau under grant GJJ12408, and Project of “Hundred Talents Plan” of CAS.

Xiong Wan has conceived and designed the study. Peng Wang, Zhimin Zhang and Huaming Zhang have partly collected and analyzed the data. The paper is mainly written by Xiong Wan.

The authors declare no conflict of interest.

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Physical Models | Errors (%) | Algorithms | ||||
---|---|---|---|---|---|---|

FE | CE | ME | ART | SIRT | ||

F_{1} | e_{1} | 0.50 | 0.56 | 0.51 | 0.59 | 1.20 |

e_{2} | 4.63 | 4.67 | 4.74 | 6.02 | 8.29 | |

e_{3} | 2.23 | 2.41 | 2.55 | 2.79 | 5.23 | |

F_{2} | e_{1} | 1.51 | 1.56 | 1.27 | 2.17 | 2.99 |

e_{2} | 11.87 | 12.17 | 14.09 | 19.02 | 27.52 | |

e_{3} | 11.84 | 13.66 | 13.32 | 19.27 | 25.75 | |

F_{3} | e_{1} | 1.23 | 1.29 | 1.42 | 2.12 | 2.73 |

e_{2} | 9.33 | 9.67 | 17.79 | 16.13 | 23.72 | |

e_{3} | 8.76 | 9.03 | 11.41 | 15.23 | 19.40 | |

F_{4} | e_{1} | 0.70 | 0.72 | 0.93 | 1.33 | 1.71 |

e_{2} | 3.29 | 3.33 | 4.87 | 8.09 | 11.17 | |

e_{3} | 3.70 | 3.80 | 4.87 | 7.33 | 9.79 |

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